The Brexit referendum question was flawed in its design by ignoring Kenneth Arrow’s impossibility theorem, writes Thomas Colignatus. As he explains, referendums can be considered democratically legitimate only if voters can make an informed decision. And it is questionable whether the UK’s referendum on the EU produced a sound choice in the first place.

Theresa May’s government, with support from the UK Parliament, has adopted Brexit as its policy aim and has invoked Article 50. Yet, economic theory assumes rational agents, and even governments might be open for rational reconsideration.

The unsatisfactory referendum question

Based upon voting theory, the Brexit referendum question can be rejected as technically unsatisfactory. One could even argue that the UK government should have annulled the outcome based on this basis alone. Even more ambitiously, one might imagine that economists and political scientists across Europe take up this issue and hence provide a basis for the EU Commission to negotiate for a proper referendum question. The big question is why the UK procedures didn’t produce a sound referendum choice in the first place.

Renwick et al. (2016) in an opinion in The Telegraph June 14 protested: ‘A referendum result is democratically legitimate only if voters can make an informed decision. Yet the level of misinformation in the current campaign is so great that democratic legitimacy is called into question’.

Their letter complains about the quality of information available to voters (an issue about which the RES has raised complaints with the BBC). It doesn’t make the point that the UK government, by ignoring voting theory, has posed a very misleading question given the complexity of the issue under decision. Quite unsettling is the Grassegger and Krogerus (2017) report about voter manipulation by Big Data, originally on Brexit and later for the election of Donald Trump. But the key point here concerns the referendum question itself.

The question assumes a binary choice — Remain or Leave the EU — while voting theory warns that allowing only two options can easily be a misleading representation of the real choice. When the true situation is more complex, and especially if it is one that arouses strong passions, then reducing the question to a binary one might suggest a political motivation. As a result of the present process, we actually don’t know how people would have voted when they had been offered the true options.

Compare the question: ‘Do you still beat your mother ?’
When you are allowed only a Yes or No answer, then you are blocked from answering:

‘I will not answer that question because if I say No then it suggests that I agree that I have beaten her in the past.’

In the case of Brexit, the hidden complexity concerned:
— Leave, and adopt an EFTA or WTO framework?
— Leave, while the UK remains intact or while it splits up?
— Remain, in what manner?

Voting theory generally suggests that representative democracy — Parliament — is better than relying on referendums, since the representatives can bargain about the complex choices involved.

Deadlocks can lurk in hiding

When there are only two options then everyone knows about the possibility of a stalemate. This means a collective indifference. There are various ways to break the deadlock: voting again, the chairperson decides, flip a coin, using the alphabet, and so on. There is a crucial distinction between voting (vote results) and deciding. When there are three options or more there can be a deadlock as well. It is less well-known that there can also be cycles. It is even less recognised that such cycles are actually a disguised form of a deadlock.

Take for example three candidates A, B and C and a particular distribution of preferences. When the vote is between A and B then A wins. We denote this as A > B. When the vote is between B and C then B wins, or B > C. When the vote is between C and A then C wins or C > A. Collectively A > B > C > A. Collectively, there is indifference. It is a key notion in voting theory that there can be distributions of preferences, such that a collective binary choice seems to result into a clear decision, while in reality there is a deadlock in hiding.

Kenneth Arrow who passed away on February 21 used these cycles to create his 1951 ‘impossibility theorem’. Indeed, if you interpret a cycle as a decision then this causes an inconsistency or an ‘impossibility’ with respect to the required transitivity of a (collective) preference ordering. However, reality is consistent and people do really make choices collectively, and thus the proper interpretation is an ‘indifference’ or deadlock. It was and is a major confusion in voting theory that Arrow’s mathematics are correct but that his own verbal interpretation was incorrect.

Representative government is better than referendums

Obviously a deadlock must be broken. Again, it may be political motivation that reduces the choice from three options A, B and C to only two. Who selects those two might take the pair that fits his or her interests. A selection in successive rounds as in France at the moment is no solution. There are ample horror scenarios when bad election designs cause minority winners. Decisions are made preferably via discussion in Parliament. Parliamentarian choice of the Prime Minister is better than direct election like for the US President.

Voting theory is not well understood in general. The UK referendum in 2011 on Alternative Vote (AV) presented a design that was far too complex. Best is that Parliament is chosen in proportional manner as in Holland, rather than in districts as in the UK or the USA. It suffices when people can vote for the party of their choice (with the national threshold of a seat), and that the professionals in Parliament use the more complex voting mechanisms (like bargaining or the Borda Fixed Point method). It is also crucial to be aware that the Trias Politica model for democracy fails and that more checks and balances are required, notably with an Economic Supreme Court.

The UK Electoral Commission goofed too

The UK Electoral Commission might be abstractly aware of this issue in voting theory, but they didn’t protest, and they only checked that the Brexit referendum question could be ‘understood’. The latter is an ambiguous notion. People might ‘understand’ quite a lot but they might not truly understand the hidden complexity and the pitfalls of voting theory. Even Nobel Prize winner Kenneth Arrow gave a problematic interpretation of his theorem. The Electoral Commission is to be praised for the effort to remove bias, where the chosen words ‘Remain’ and ‘Leave’ are neutral, and where both statements were included and not only one. (Some people don’t want to say ‘No’. Some don’t want to say ‘Yes’.) Still, the Commission gives an interpretation of the ‘intelligibility’ of the question that doesn’t square with voting theory and that doesn’t protect the electorate from a voting disaster.

A test on this issue involves asking yourself: Given the referendum outcome, do you really think that the UK population is clear in its position, whatever the issues of how to leave or the risk of a UK breakup? If you have doubts on the latter, then you agree that something is amiss. The outcome of the referendum really doesn’t give us a clue as to what UK voters want. Scotland wants to remain in the EU and then break up? This is okay for the others who want to Leave? (And how?) The issue can be seen as a statistical enquiry into what views people have, and the referendum question is biased and cannot be used for sound conclusions.

In an email to the author in July 2016 a spokesman for the Electoral Commission said its role: ‘… is to evaluate the intelligibility of referendum questions in line with the intent of Parliament; it is not to re-evaluate the premise of the question. Other than that, I don’t believe there is anything I can usefully add to our previously published statements on this matter.’

Apparently the Commission knows the ‘intent of Parliament’, while Parliament itself might not do so. Is the Commission only a facilitator of deception, and don’t they have a mission to put voters first? At best the Commission holds that Whitehall and Parliament fully understood voting theory and therefore intentionally presented the UK population with a biased choice, so that voters would be compelled to neglect the complexities of leaving or even a break-up of the Union. Obviously the assumption that Whitehall and Parliament fully grasp voting theory is dubious. The better response by the Commission would have been to explain the pitfalls of voting theory and the misleading character of the referendum question, rather than facilitate the voting disaster.

Any recognition that something is (very) wrong here, should also imply the annulment of the Brexit referendum outcome. Subsequently, to protect voters from such manipulation by Whitehall, one may think of a law that gives the Electoral Commission the right to veto a biased Yes / No selection, which veto might be overruled by a 2/3 majority in Parliament. Best is not to have referendums at all, unless you are really sure that a coin can only fall either way, and not land on its side (by a hidden deadlock).

19 Comments

I simply must vote Leave as this was the democratic wish of the UK referendum.
Whilst I as a true upstanding member of the UK I must stand firm on supporting our democratic principals and that me and vote to leave despite voting remain.

I heard recently that the referendum about remaining or leaving the EU had only a simple, rather than a super majority because it was intended as advisory only.
Has anyone any information about whether this is correct?
And if it is, why did the government immediately treat it as binding?

We haven’t had many referenda in the UK. None has ever required a ‘super’ majority. The previous referendum on remaining in the EU was worded almost identically, was also held after a dishonest campaign on both sides, also required only a ‘simple’ majority, was also, technically, advisory only, and also changed our constitution – much more than was then admitted by the politicians – and has been accepted since 1975 as legitimate. The objections to this later one only appeared after the remain campaign – which was sure it had the result in the bag – was proved wrong by the result.

I totally agree with Thomas Colignatus: ‘Renwick et al. (2016) in an opinion in The Telegraph June 14 protested: ‘A referendum result is democratically legitimate only if voters can make an informed decision. Yet the level of misinformation in the current campaign is so great that democratic legitimacy is called into question’.

Yes! Exactly what I, and millions of others in Britain, have been saying ever since. It was/is utter madness to base such a cataclysmic change in the country’s fortunes and future, based on the vote of millions of people who either didn’t know the facts and possible consequences of what they were voting for – or chose to ignore or dismiss the wise words of those who tried to warn them – dismissing these people (derisively!) as “experts!” I mean, Jesus, if you’re going to reject the words of people who know better… you are surely wilfully embracing ignorance! And the screaming headlines of the biased, predominantly Right-wing Little England Press, like The Sun, Mail,Telegraph and Express (with its disgraceful daily anti-immigrant hectoring…), only fuelled the anti-Europe hysteria.

So… you had a large, ignorant, blinkered, or pig-headed proportion of the electorate…and then an outcome divided only approx 48/52…. and on that slender basis, (surely no move should have been contemplated below at least a 40/60 split…), we are pulling away from Europe to a future unknown…

And for May and others to keep chanting, “It’s the will of the people…,” No, it isn’t…! It isn’t!!! It’s the will of less than 50% (allowing for the proportion that voted). The rest of us, the majority (!!), are being dragged over this cliff by a vocal minority, a biased press – and by a disappointingly spineless leader (Theresa May), who hasn’t the balls to halt this madness, and review the events and nature of this debacle that she inherited. Far from leading, she is following the unenlightened herd… rather than stand up against the rabid anti-Europeans in her party, as Cameron failed to do.

Why the silence on the legitimacy or otherwise of the 1975 referendum? On the duplicity of the ‘yes’ campaign then? On the uncertainty inbuilt into its design – in that the areas of sovereignty that would be ‘pooled’ (or rather, ceded) to the supranational organs of the EU and the import of joining, for example, the Coal and Steel Union, the Common Agricultural Policy and the Common Fisheries Policy were not honestly spelled out? The UK has lived with the consequences of those deceits for over 40 years

Where was the outrage when the 2016 referendum was announced? When its question was published? When the BBC giggled about its triviality and how boring it was prior to the vote itself, and persistently returned their coverage to ‘things that really matter to people’ – like ‘Bake-Off’? The truth is that those who have now discovered how dishonest the ‘leave’ campaign was (but apparently not the lies of the ‘remain’ campaign) have only become outraged because the result of the referendum was not what they assumed it would be.

“The EU is illiberable, protectionist and undemocratic. It will ultimately fail. The future lies in global free trade under the rule of law. Britain will lead the way.”

Where, I wonder, will the US, China, India plus stand in your suggested world under the rule of law? You believe, don’t you?, that they will all agree to follow you, the global-minded Brits…I ow have strong feeling that it would be much better for EU to have UK leave. You shouldn’t have joined it in 1973 in the first place.

Whilst accepting that referenda are very poor mechanisms for policy determination – and that disinformation on both sides was rife during the Brexit Referendum – you seem to have missed the crucial point; that it delivered the right result!

Our liberty was threatened by rules from the ECJ – a court constituted to rule always in the interests of “Ever Closer Union” rather than blind justice before the law. Our trade was threatened by protectionism that sought to enmesh us in a market that, having already diminished from 73% of our trade (1973) to 40% (2016 taking into account the Rotterdam effect), will inevitably because of global demographic trends, shrink to relative insignificance. Our productivity and the incentives to automate our agriculture and service industries have been damaged by the enforced influx of artificially cheap labour. Our ability to maintain living standards by expenditure on new infrastructurr has been overwhelmed by uncontrolled immigration. Bien pensant liberals do not care about the cultural dissonance created by our failure to build ten citiies the size of Cardiff in the last ten years.

The EU broke its own rules to permit accession by countries that were non-compliant. The Euro was established before fiscal union had been achieved – and then southern European countries were allowed to join the Euro when they had failed to achieve the required economic performance. The permanent immiseration of the Southern European countries cannot be tolerated and will lead to the failure of the Euro.

The EU is illiberable, protectionist and undemocratic. It will ultimately fail. The future lies in global free trade under the rule of law. Britain will lead the way.

The “Right Result,” Nigel Holder, in your view. Numerous others, like me, believe it was absolutely the wrong result…, based on an insubstantial, and ridiculously flawed vote, as argued here. It really wasn’t and isn’t “The Will of the People…!!”

Thanks for sharing your thoughts. Fully agree on all points you made. A few months prior to referendum, I wrote a piece where I pointed to the same shortcomings: referendum not a right format for such a decision; the issue itself narrow-framed thus not allowing for a strategic perspective; as a result, the question is wrongly formulated; and the pre-referendum discourse does not contribute to better understanding of the problem at hand, by citizens. I called it a political communication failure. Independently of the outcome, the entire affair was handled very unprofessionally by the British politicians. That is why it is so broadly and fiercely contested; it should not have been like that.

That you think the UK had a referendum on proportional representation in 2011 – when we did not – suggests that you do not know much about voting theory. I trust it’s just a mistake, but your arguments are thus undermined.

Voters are supposed to do their own due diligence. Who is supposed to give them an informed choice? A government that wants to remain, the EU?

We had 2 questions to get into the EEC back in 1975 and *everyone* would have had at least 18 years experience of the EU and quite frankly years of propaganda from both sides to make their choice in the referendum last summer over 40 years on from 1975.

The Electoral Commission couldn’t impose any rules without agreement from both sides – and there is no way Brexiteers and even all of the Tories would agree to referendum questions that has the potential to undermine everything by turning into a Scottish Indy2 referendum.

The UK cannot “split up” under any Brexit because Westminster won’t have any of that until Brexit is in the distance behind us.

You still fail to understand a lot of the reasons why so many voted for Brexit in the first place.

Only one Treaty, the Treaty of Maastrict was ever in a manifesto with a proper mandate for the government to sign.

With the rest there was no mandate and even worse, the Treaty of Lisbon which is really the European Constitution – we was promised a referendum by Labour. And they stuck their two fingers up at the electorate once in power.

I notice many Remainers are still maligning the the public’s view on the issue of the single market, by claiming that just because people want access to the single market, that they somehow want the country to remain members of it.

Also, I find it breathtaking that you believe we should be getting direction off the European Commission on how we should go about something so basic as to what question we should ask our own citizens in such a referendum. They are not the final arbiters in this, we don’t need their approval, even a government that intends to remain, would be a laughing stock if we somehow deferred to the European Commission on the wording of such a referendum.

It’s well known that the referendum was and is illegitimate. This fact is ignored and denied by everyone in the country and most of the EU. The gross delusion displayed by the government and parliament is curious. The catastrophic disaster of Brexit will come to pass and yet the UK will continue in denial and lies while the world accepts the UK as a tiny irrelevant nation not worth serious consideration other than a great big laughing stock. Perhaps the young generations will figure it out by and for themselves and come to forgive the Brexwits and the entire UK population for throwing away the sacrifices made in the 1914-1945 war. Manchester is pretending to unite and express the Remain slogan of “stronger together” while the government and T. May uses security as a bargaining pawn. We live in comically and ludicrously stupid times. The farce continues.

Jacob Jonker, you can question Mozartsbum’s use of the word “illegitimate,” if you want, but, surely he agrees with the tenet of the piece: that the Brexit Referendum was so clearly and self-evidently flawed, that it is bizarre, bewildering, and frustrating to those of us with a brain and sense, that its mechanism and outcome is not being universally questioned…

There are alarming echoes of “1984” in all this… that we should all accept without question, the state-recognised “Good…,” and shout down, silence, and bury those who try to argue otherwise…

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